https://doi.org/10.1051/cocv/2020060 www.esaim-cocv.org
MULTIOBJECTIVE OPTIMAL CONTROL OF A NON-SMOOTH SEMILINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION
∗Constantin Christof
1,∗∗and Georg M¨ uller
2Abstract. This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are – in a certain sense – exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures.
Mathematics Subject Classification. 35J20, 49J52, 49K20, 58E17, 90C29.
Received January 27, 2020. Accepted September 19, 2020.
∗This research has been partially funded by the German Research Foundation (DFG) through the Priority Programme SPP 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimiza- tion”, Project P02 “Multiobjective Optimization of Non-smooth PDE-constrained Problems – Switches, State Constraints, and Model Order Reduction”. The first author gratefully acknowledges the support by the International Research Training Group IGDK 1754, funded by the German Research Foundation (DFG) and the Austrian Science Fund (FWF) under project number 188264188/GRK1754.
Keywords and phrases: Multiobjective optimal control, non-smooth optimization, first-order necessary optimality condition, strong stationarity, semilinear partial differential equation, Pareto front.
1 Technische Universit¨at M¨unchen, Chair of Optimal Control, Center for Mathematical Sciences, M17, Boltzmannstraße 3, 85748 Garching, Germany.
2 Universit¨at Konstanz, Department of Mathematics and Statistics, WG Numerical Optimization, Universit¨atsstraße 10, 78457 Konstanz, Germany.
*∗Corresponding author:[email protected]
Article published by EDP Sciences c EDP Sciences, SMAI 2021
1. Introduction
The aim of this paper is to study first-order necessary optimality conditions for multiobjective optimal control problems of the form
Minimize
J1(y, u) :=j1(y) +ν1
2 kuk2L2
...
JN(y, u) :=jN(y) +νN 2 kuk2L2
w.r.t. u∈L2(Ω), y∈H01(Ω)∩H2(Ω), s.t. −∆y+ max(0, y) =ua.e. in Ω.
(P)
Here, Ω⊂Rd, d≥1, is a bounded domain with a sufficiently regular boundary, νn, n= 1, . . . , N, are non- negative Tikhonov parameters with νN >0, and jn, n= 1, . . . , N, are given objective functions with suitable mapping and smoothness properties. For the precise assumptions on the quantities in (P), we refer to Section2.
Problems of the type (P) – i.e., multicriteria optimization problems governed by semilinear partial differential equations involving non-smooth Nemytskii operators - arise, for instance, in mechanics, plasma physics, and the context of certain combustion processes when the state of the system is supposed to meet several, potentially conflicting design goals. See, e.g., [33,47,53,55] for some examples of possible application areas.
From the mathematical point of view, problems of the type (P) are challenging for a number of reasons. The first and probably most obvious one is the presence of the non-smooth Nemytskii operator in the governing partial differential equation. Because of this term, the control-to-state mapping S : u7→y of (P) does not possess a Gˆateaux derivative, but is only directionally differentiable, and it is not possible to apply standard results to derive, for instance, first-order necessary optimality conditions or to devise efficient numerical solution algorithms. The strategy that is most commonly used in the literature to handle such a lack of smoothness is to work with elements of appropriately defined subdifferentials (e.g., those of Clarke, Dini, Mordukhovich, or Fr´echet) instead of the non-existing gradients of the involved (reduced) objective functions. See, for example, [17, 23, 43, 50, 52] for an overview of these concepts and references on the use of subgradients in the single- objective context, [9,10,26,27,30,31,36,39,42,56] for applications in the multiobjective setting, and [3,6,35]
for results that additionally also rely on regularization techniques. Unfortunately, for problems of the type (P), a standard subgradient-based analysis turns out to be not very rewarding either. Since the non-smoothness enters (P) only indirectlyvia the non-differentiable Nemytskii operator in the governing PDE, a full characterization of quantities like the Clarke subdifferentials of the reduced objective functions Jn(S(·),·), n = 1, . . . , N, of (P) is only available in pathological situations. Optimality conditions involving these generalized differentials are thus rather academic and barely usable in practice. Note that this is a major difference to situations, in which the non-smoothness stems from the “outer” functionsJn and not from the governing PDE, and in which, as a consequence, classical chain rules for subgradients can be applied, cf. [7]. A possible way around this difficulty, that has recently been employed in [14,48,49] in the single-objective setting, is to work with notions of generalized derivatives on the level of the solution operator S :u7→ y. Such approaches typically result in necessary optimality conditions of intermediate strength that can indeed be solved with standard solution algorithms. For more details on this topic and further remarks on how the findings of [14, 48, 49] are related to the present paper, we refer to [14], Section 4 and Sections 4 and 5. For results on smooth multiobjective optimization and optimal control problems, see also [4,8,22,29, 44–46].
A second factor that significantly complicates the derivation of necessary optimality conditions for problems of the type (P) and that is completely absent in the single-objective setting is that - even in the smooth case – there are several sensible, purely primal optimality and stationarity concepts for multiobjective optimization problems.
Compare, for instance, with the notions of weak, ordinary, and proper Pareto optimality and stationarity in Definitions 2.3 and 3.2in this context. Even worse, the multiobjective aspect of (P) also turns out to add an additional layer of non-smoothness to the problem. To see this, consider, for example, the notion of weak Pareto
optimality for (P),i.e., the optimality condition given by
@u∈L2(Ω) : Jn(S(u), u)< Jn(S(¯u),u)¯ ∀n= 1, . . . , N.
This condition is clearly equivalent to ¯ubeing a global optimum of the scalar problem min
u∈L2(Ω)max J1(S(u), u)−J1(S(¯u),u), . . . , J¯ N(S(u), u)−JN(S(¯u),u)¯
, (1.1)
whose objective function not only contains the non-smooth solution map S, but also the non-differentiable maximum-value function on RN. We thus indeed have to deal with two sources of non-smoothness here which are even nested. Note that, in first-order necessary optimality conditions for weak Pareto optima, the presence of the maximum-value function in (1.1) typically manifests itself in the form of additional multipliers. Compare, for instance, with the results for multiobjective problems with smooth objective functions in [39], Theorems 3.1.1, 3.1.5 in this context and also with the stationarity systems established in Theorem 4.5.
The main goal of the present paper is to demonstrate that, despite all of the above difficulties, it is indeed possible to derive very rigorous first-order necessary optimality conditions for multiobjective optimal control problems of the type (P). To be more precise, we will show how to prove so-called strong stationarity conditions for this problem class. In the single-objective setting, such conditions are well-known, e.g., for the optimal control of elliptic variational inequalities of the first and the second kind and the optimal control of non-smooth elliptic and parabolic partial differential equations, see [13,14,18,19,38,40,41]. In the multiobjective context, strong stationarity conditions have, at least to the best of the authors’ knowledge and with some exceptions for finite-dimensional MPECs in [56], not been considered so far (most likely because of the doubly non-smooth behavior in (1.1) that is generally hard to handle). Recall that the distinguishing feature of a strong stationarity system is its equivalence to the first-order necessary optimality condition in primal form. In the single-objective case, this means that a point is strongly stationary if and only if it is Bouligand stationary in the sense of [20], Definition 5.4,i.e., stationary in the sense that the directional derivative is non-negative in all directions.
When considering the problem (P) with its multiple sensible purely primal necessary optimality conditions, one, of course, has to differentiate at this point. We will thus establish not only one but even two strong stationarity systems for (P) – one equivalent to weak Pareto stationarity and one equivalent to both ordinary and proper Pareto stationarity, see Definition 3.2and Theorem4.5. (Note that this implies in particular that the concepts of ordinary and proper Pareto stationarity are the same for the problem (P).) A main ingredient of our analysis is the - at first glance rather surprising – fact that the non-linear and non-differentiable inverse S0(u;·)−1:H01(Ω)∩H2(Ω)→L2(Ω),z7→ −∆z+1{S(u)=0}max(0, z) +1{S(u)>0}z, of the directional derivative S0(u;·) of the control-to-state mappingS:u7→y of (P) is self-adjoint in the sense that
u, S0(u;·)−1(z)
L2=
S0(u;·)−1(u), z
H01∩H2 (1.2)
holds for allu∈L2(Ω) and allz∈H01(Ω)∩H2(Ω), where, on the right-hand side of (1.2), the mapS0(u;·)−1 is interpreted as a function from L2(Ω) to (H01(Ω)∩H2(Ω))∗. This behavior makes it possible to resolve the difficulties related to the additional layer of non-smoothness in (1.1) that normally prevent the derivation of strong stationarity conditions in the multiobjective context, cf.Lemma4.2and Theorem4.5.
Note that the self-adjointness of the operator S0(u;·)−1 in (1.2) is also of relevance for scalar Tikhonov regularized optimal control problems governed by non-smooth semilinear partial differential equations as it allows to employ an adjoint calculus similar to that available in the smooth setting in the non-smooth case. Quite interestingly, the self-adjointness property in (1.2) is also directly related to certain non-standard regularization effects that have apparently not been documented so far in the literature – neither in the single- nor in the multiobjective context. These effects cause the problem (P) to be Gˆateaux differentiable when it is reduced to the state y although it is not Gˆateaux when reduced to the controlu. For further details on this topic, see Section 6. We remark that the results of Section 6 particularly imply that the problems considered in [18],
Section 5, [14], Section 5, and [21], Section 5, Case 1 all admit a Gˆateaux differentiable reformulation. As we will see in Section5, they further yield that regularization approaches are exact for the problem (P) in the sense that weakL2-accumulation points of weakly Pareto stationary points of the regularized multiobjective optimal control problems are weakly Pareto stationary for the non-smooth limit problem. Note that this behavior is again quite surprising as such effects can typically not even be observed in simple, one-dimensional examples.
Compare, for instance, with the situation, where the function f(x) :=−|x| is approximated by the sequence fε(x) :=−√
x2+ε, ε >0, and where the point ¯x= 0 is Bouligand stationary for allfε but not for the limit functionf, in this context.
Before we begin with our analysis, we would like to point out that we consider the problem (P) as a model problem in this paper. It is easy to check that our arguments can be extended straightforwardly to cases where the governing PDE contains a more general elliptic second-order partial differential operator or a more complicated piecewise C1-function with properties similar to that of max(0,·). An extension to the parabolic setting,cf.[38], is also possible by invoking the results in the appendix of [15]. We restrict our attention to the setting in (P) to avoid obscuring the basic ideas of our analysis with unnecessary technicalities.
To help the reader navigate the paper, we conclude this introduction with a brief overview of the content of the upcoming sections:
Sections 1.1 and 2 deal with preliminaries. Here, we comment on the notation used, state our standing assumptions onjn,νn, and Ω (see Asm.2.1), and recall known results on the properties of the control-to-state mapping S:u7→y of (P), notions of optimality for multiobjective optimization problems, and the existence of Pareto optimal controls.
Section3is concerned with first-order necessary optimality conditions in purely primal form (i.e., conditions based on directional derivatives). We remark that, in the finite-dimensional setting, such stationarity concepts have already been discussed,e.g., in [26], Section 4 and [30], Section 3.
In Section 4, we prove the already mentioned strong stationarity conditions for the multiobjective optimal control problem (P). This section contains the main result of the paper, Theorem 4.5. It further addresses in detail the self-adjointness property in (1.2) (see Lem. 4.2), the consequences that our findings have for the regularity properties of Pareto stationary points (see Cor.4.7), and the relationship of our results to the notion of subdifferential studied in [14,48,49] and classical scalarization techniques (see Rem. 4.6).
Section 5 demonstrates that smoothing methods are indeed exact for problems of the type (P) in the sense that they allow to determine weakly Pareto stationary points when the regularization parameter is driven to zero, see Theorem5.3. Corollary5.4in this section moreover shows that the concepts of strong and C-stationarity are the same for the problem (P) in the single-objective caseN = 1.
In Section 6, we discuss the non-standard regularization effects that are responsible for the results of the previous sections and that cause the problem (P) to be Gˆateaux differentiable when reduced to the statey. Here, we further give an alternative interpretation of the strong stationarity conditions in Theorem4.5and prove an auxiliary result on non-smooth Nemytskii operators that is also interesting on its own, see Theorem6.1.
Section7of the paper finally contains numerical experiments which demonstrate that the multiplier systems in Sections4and5are amenable to numerical solution procedures and allow to compute approximations of the Pareto front of (P).
1.1. Remarks on the notation
In what follows, we use the standard symbols Lq(Ω), Ck,γ(Ω), H0k(Ω),Hk(Ω), and Wk,q(Ω), 1 ≤q ≤ ∞, k∈N, 0< γ≤1, to denote the Lebesgue-, H¨older-, and Sobolev spaces on a bounded Lipschitz domain Ω⊂Rd, d≥1, respectively. For details on these spaces, we refer to [1,2,24,28]. Given two Banach spacesX andY, we further defineX∗to be the dual space ofX andL(X, Y) to be the space of linear and continuous functions from X toY. In the special caseX =H01(Ω), we also setH−1(Ω) :=H01(Ω)∗. As usual, we interpretH01(Ω),L2(Ω), and H−1(Ω) as a Gelfand triple,i.e.,H01(Ω),→L2(Ω)∼=L2(Ω)∗ ,→H−1(Ω). The same convention is used for the space H01(Ω)∩H2(Ω),cf. [32], Section 1.9. Norms, scalar products, and dual pairings are denoted by the symbolsk · k, (·,·), andh·,·iin this paper, and the modes of weak and strong convergence by the arrows*and
→. If we want to specify the space/topology that we are referring to, then we add suitable sub- or superscripts and write,e.g.,k · kL2. With ∆,∇,∂c, and∂i,i= 1, . . . , d, we denote the (distributional) Laplacian, the (weak) gradient, the convex subdifferential, and the first (weak) partial derivatives of a function, respectively. For higher-order derivatives, we also use the multi-index notation ∂α, α∈Nd0. Gˆateaux-, Fr´echet-, and directional derivatives in the functional analytic sense are denoted by a prime in the usual way. Given a functionv: Ω→R and a measurable set D⊂Ω, we finally define{v∗0},∗ ∈ {=,6=, <, >,≤,≥}, to be the set{x∈Ω|v(x)∗0}
and1Dto be the indicator function ofD(with values in{0,1}). Ifvis an element of anLq-space andDis only defined up to sets of measure zero, then we consider{v∗0}to be defined up to sets of measure zero as well and identify 1D with an element ofL∞(Ω).
Note that, in the remainder of this paper, new symbolsetc.are introduced whenever necessary. For the sake of readability, this additional notation is defined where it first appears in the text.
2. Problem setting and preliminaries
As already mentioned in the introduction, the aim of this paper is to study multiobjective optimal control problems of the type
Minimize
J1(y, u) :=j1(y) +ν1 2 kuk2L2
...
JN(y, u) :=jN(y) +νN
2 kuk2L2
w.r.t. u∈L2(Ω), y∈H01(Ω)∩H2(Ω), s.t. −∆y+ max(0, y) =ua.e. in Ω.
(P)
Our standing assumptions on the quantities in (P) are as follows:
Assumption 2.1 (Standing assumptions for the study of problem (P)).
i) Ω⊂Rd, d≥1, is a bounded domain that is convex or possesses a C1,1-boundary (in the sense of [25], Sect. 6.2).
ii) jn :H01(Ω)∩H2(Ω) → R, n= 1, . . . , N, N ∈ N, are functions that are weakly lower semicontinuous, continuously differentiable, and bounded from below.
iii) νn,n= 1, . . . , N, are given non-negative real numbers andνN is positive.
Note that the PDE in (P) is uniquely solvable for allu∈L2(Ω) by the theorem of Browder and Minty, [51], Theorem 3-1.5. To be more precise, we have:
Proposition 2.2 (Properties of the PDE in (P)). For every right-hand side u∈L2(Ω), there exists a unique solution y∈H01(Ω)∩H2(Ω) of the partial differential equation
−∆y+ max(0, y) =ua.e. inΩ. (2.1)
Further, the solution operator S:L2(Ω)→H01(Ω)∩H2(Ω),u7→y, associated with the PDE (2.1) satisfies:
i) S is globally Lipschitz continuous, i.e., there exists an absolute constantC >0 with kS(u1)−S(u2)kH2 ≤Cku1−u2kL2 ∀u1, u2∈L2(Ω).
ii) S is weakly continuous, i.e., for every u∈L2(Ω)it holds uk
L2
* u =⇒ S(uk)H
2
* S(u). (2.2)
iii) S is strongly and weakly Hadamard directionally differentiable in every point u∈L2(Ω) in every direc- tion v∈L2(Ω), i.e., for all u, v ∈L2(Ω), there exists a unique S0(u;v)∈H01(Ω)∩H2(Ω) such that the implications
vk L2
* v, tk→0+ =⇒ S(u+tkvk)−S(u) tk
H2
* S0(u;v) and
vk→L2v, tk→0+ =⇒ S(u+tkvk)−S(u) tk
H2
→S0(u;v)
hold. Moreover, the directional derivativeδv:=S0(u;v)∈H01(Ω)∩H2(Ω)in a pointuwith statey:=S(u) in a directionv is uniquely characterized by the partial differential equation
−∆δv+1{y=0}max(0, δv) +1{y>0}δv=v a.e. inΩ. (2.3) iv) S is Gˆateaux differentiable in a point u∈L2(Ω) with state y :=S(u) (i.e., S0(u;·) is an element of the
spaceL(L2(Ω), H01(Ω)∩H2(Ω))) if and only if the set{y= 0} has measure zero.
Proof. All assertions of the proposition have been proved in [14], Proposition 2.1, Theorem 2.2, Corollary 2.3, 3.8.
Note that, under our assumptions on Ω, the space (Y,k · kY) used in [14] is isomorphic to (H01(Ω)∩H2(Ω),k · kH2) by [25], Theorem 9.15, Lemma 9.17 and [28], Theorem 3.2.1.2. The results of [14] thus indeed yield the asserted mapping properties of the solution operator S.
As usual in the context of multiobjective optimization, in the remainder of this paper, we are interested in finding controls u∈L2(Ω) that yield – at least in some sense – an optimal compromise between the different objective functions Jn(S(·),·), n= 1, . . . , N, of (P). The notions of optimality that we are mainly concerned with in our analysis are the following (cf.[30], Defs. 3.1, 3.2 and also [22,39]):
Definition 2.3 (Notions of Pareto optimality). A control ¯u∈L2(Ω) with associated state ¯y:=S(¯u) is called:
i) a local weak Pareto optimum of (P) if there exists anr >0 such that there is nou∈L2(Ω) satisfying ku−uk¯ L2 < r, Jn(S(u), u)< Jn(¯y,u)¯ ∀n= 1, . . . , N.
ii) a local Pareto optimum of (P) (in the ordinary sense) if there exists an r > 0 such that there is no u∈L2(Ω) satisfying
ku−uk¯ L2 < r, Jn(S(u), u)≤Jn(¯y,u)¯ ∀n= 1, . . . , N, Jn(S(u), u)< Jn(¯y,u) for at least one¯ n∈ {1, . . . , N}.
iii) a local proper Pareto optimum of (P) (in the sense of Geoffrion) if there exist constantsr, C >0 such that, for every controlu∈L2(Ω) satisfyingku−uk¯ L2 < r and Jl(S(u), u)< Jl(¯y,u) for some¯ l∈ {1, . . . , N}, there exists an indexm∈ {1, . . . , N} with
Jl(¯y,u)¯ −Jl(S(u), u)≤C Jm(S(u), u)−Jm(¯y,u)¯ .
iv) a global weak/ordinary/proper Pareto optimum, respectively, of (P) if the condition in i)/ii)/iii), respectively, holds withr=∞.
Note that we trivially have
¯
uproperly Pareto optimal ⇒ u¯ Pareto optimal ⇒ u¯weakly Pareto optimal, (2.4) and that the concepts of local (respectively, global) weak, ordinary, and proper Pareto optimality coincide with each other and the classical notion of local (respectively, global) optimality in the single-objective case N = 1.
Using our standing assumptions and the properties of the map S in Proposition2.2, it is easy to prove:
Theorem 2.4 (Existence of proper Pareto optima). There exists at least one global proper Pareto optimum
¯
u∈L2(Ω) of the problem (P).
Proof. We use a scalarization approach,cf.[22], Theorem 3.11 and [39]: Consider the auxiliary problem
u∈Lmin2(Ω) N
X
n=1
jn(S(u)) +νn 2 kuk2L2
. (2.5)
Then, it follows from our assumptions onjnandνn, the weak continuity of the mapSin Proposition2.2ii), and the weak lower semicontinuity of convex and continuous functions that the objective of (2.5) is weakly lower semicontinuous, bounded from below, and radially unbounded as a function fromL2(Ω) toR. These properties imply, in combination with the direct method of calculus of variations, that (2.5) admits at least one global minimum ¯u∈L2(Ω),i.e., at least one ¯u∈L2(Ω) with associated state ¯y:=S(¯u) such that
N
X
n=1
Jn(¯y,u)¯ ≤
N
X
n=1
Jn(S(u), u) ∀u∈L2(Ω). (2.6)
We claim that this ¯uis also a global proper Pareto optimum of the problem (P). To see this, suppose that we are given a u∈L2(Ω) that satisfiesJl(S(u), u)< Jl(¯y,u) for some¯ l∈ {1, . . . , N}. Then, (2.6) yields
0< Jl(¯y,u)¯ −Jl(S(u), u)≤X
n6=l
Jn(S(u), u)−Jn(¯y,u)¯
≤N max
n=1,...,N Jn(S(u), u)−Jn(¯y,u)¯
, (2.7)
and we may deduce that there exists anm∈ {1, . . . , N} with
Jl(¯y,u)¯ −Jl(S(u), u)≤N Jm(S(u), u)−Jm(¯y,u)¯ .
This shows that the condition in Definition2.3iii)holds (withr=∞andC=N) and completes the proof.
3. First-order necessary optimality conditions in primal form
Having discussed the properties of the PDE (2.1) and the solvability of the problem (P), we now turn our attention to first-order necessary optimality conditions. We begin with “purely primal” optimality conditions that rely only on directional derivatives and do not involve additional multipliers. In the finite-dimensional setting, such conditions have already been discussed, for instance, in [26], Section 4 and [30], Section 3.
Theorem 3.1 (First-order necessary optimality conditions in primal form).
i) Ifu¯∈L2(Ω) is a local weak Pareto optimum of (P) with associated state y¯:=S(¯u), then there exists no v∈L2(Ω)satisfying
hjn0(¯y), S0(¯u;v)iH1
0∩H2+νn(¯u, v)L2 <0 ∀n= 1, . . . , N. (3.1)
ii) If u¯∈L2(Ω) is a local proper Pareto optimum of (P) with state y¯:=S(¯u) and constants r, C >0 as in Definition2.3iii), then, for every directionv∈L2(Ω)satisfyinghj0l(¯y), S0(¯u;v)iH1
0∩H2+νl(¯u, v)L2 <0for somel∈ {1, . . . , N}, there exists an indexm∈ {1, . . . , N}with
−
hjl0(¯y), S0(¯u;v)iH1
0∩H2+νl(¯u, v)L2
≤C
hjm0 (¯y), S0(¯u;v)iH1
0∩H2+νm(¯u, v)L2
. (3.2) Proof. From our assumptions on the functions jn, the differentiability properties of the solution map S in Proposition 2.2, and the chain rule, [11], Proposition 2.47, it follows straightforwardly that
lim
t→0+
Jn(S(u+tv), u+tv)−Jn(S(u), u)
t =hjn0(S(u)), S0(u;v)iH1
0∩H2+νn(u, v)L2 (3.3) holds for all u, v ∈L2(Ω) and alln= 1, . . . , N. Suppose now that we are given a local weak Pareto optimum
¯
u∈L2(Ω) of (P) such that there exists a direction v∈L2(Ω) with (3.1). Then, (3.3) yields that we can find arbitrarily small numberst >0 withJn(S(¯u+tv),u¯+tv)−Jn(S(¯u),u)¯ <0 for alln= 1, . . . , N. This contradicts the local weak Pareto optimality of ¯u, shows that every local weak Pareto optimum has to satisfy the condition ini), and proves the first part of the theorem. To establishii), we can proceed along similar lines: If we are given a local proper Pareto optimum ¯u∈L2(Ω) with state ¯y:=S(¯u) and constants r, C >0 as in Definition2.3iii) and a v ∈L2(Ω) satisfying hjl0(¯y), S0(¯u;v)iH1
0∩H2+νl(¯u, v)L2 <0 for some l∈ {1, . . . , N}, then (3.3) yields that Jl(S(¯u+tv),u¯+tv)−Jl(¯y,u)¯ <0 holds for all sufficiently small t >0, and we may use the condition in Definition2.3iii)to deduce that there exist anm∈ {1, . . . , N} and a sequence{tk} ⊂R+with tk→0+ and
0< Jl(¯y,u)¯ −Jl(S(¯u+tkv),u¯+tkv)≤C Jm(S(¯u+tkv),u¯+tkv)−Jm(¯y,u)¯ for allk∈N. Due to (3.3) and the properties ofl, the above implies
0<−
hjl0(¯y), S0(¯u;v)iH1
0∩H2+νl(¯u, v)L2
≤C
hjm0 (¯y), S0(¯u;v)iH1
0∩H2+νm(¯u, v)L2
.
This establishes (3.2) and completes the proof.
Theorem3.1 motivates the following definition:
Definition 3.2 (Notions of stationarity). A control ¯u∈L2(Ω) with associated state ¯y:=S(¯u) is called:
i) a weakly Pareto stationary point of (P) if there is nov∈L2(Ω) satisfying hj0n(¯y), S0(¯u;v)iH1
0∩H2+νn(¯u, v)L2 <0 ∀n= 1, . . . , N. (3.4) ii) a Pareto stationary point of (P) if there is nov∈L2(Ω) satisfying
hj0n(¯y), S0(¯u;v)iH1
0∩H2+νn(¯u, v)L2 ≤0 ∀n= 1, . . . , N, hjn0(¯y), S0(¯u;v)iH1
0∩H2+νn(¯u, v)L2 <0 for at least onen∈ {1, . . . , N}. (3.5) iii) a properly Pareto stationary point of (P) if there exists aC >0 such that, for every v∈L2(Ω) satisfying
hjl0(¯y), S0(¯u;v)iH1
0∩H2+νl(¯u, v)L2<0 for somel∈ {1, . . . , N}, there is an index m∈ {1, . . . , N}with
−
hjl0(¯y), S0(¯u;v)iH1
0∩H2+νl(¯u, v)L2
≤C
hjm0 (¯y), S0(¯u;v)iH1
0∩H2+νm(¯u, v)L2
. (3.6)
Some remarks are in order regarding the concepts in Definition 3.2:
Remark 3.3. i) Pareto optima satisfying the condition in Definition3.2ii)are sometimes also referred to as
“efficient in Kuhn and Tucker’s sense” or “KT-proper”, see [22], Definition 2.49 and [30], Definition 3.3.
ii) Analogously to the hierarchy in (2.4), we have
¯
uproperly stationary ⇒ u¯ stationary ⇒ u¯weakly stationary.
iii) The conditions i), ii), and iii) in Definition 3.2 are equivalent to the global weak, global ordinary, and global proper Pareto optimality of the function ¯v= 0 in the first-order approximation of (P) defined by
Minimize
J1(¯y,u) +¯ hj10(¯y), S0(¯u;v)iH1
0∩H2+ν1(¯u, v)L2
... JN(¯y,u) +¯ hjN0 (¯y), S0(¯u;v)iH1
0∩H2+νN(¯u, v)L2
w.r.t. v∈L2(Ω),
respectively. This shows in particular that the concepts in Definition3.2extend the notion of Bouligand stationarity, see [20], Definition 5.4, to the multiobjective setting. (It is easy to check that, in the single- objective caseN = 1, all of the conditions in Definition3.2coincide with each other and with the notion of Bouligand stationarity.)
iv) While the conditionsi)andiii)in Definition3.2are always necessary for weak and proper Pareto optimality, respectively, by Theorem3.1, the property in Definition3.2ii)is typically not a necessary condition for ordi- nary Pareto optimality. Compare,e.g., with the simple bi-criterial optimization problem min(f1(x), f2(x)) withf1(x) :=−xandf2(x) :=x3 in this context, where the point ¯x:= 0 is a Pareto optimum but does not satisfyii). In the finite-dimensional setting, the necessity ofii) can be recovered under a generalized Abadie constraint qualification, see [26], Theorem 4.1.
Due to their reliance on the non-linear directional derivative S0(¯u;·) of the solution map S and their for- mulation as variational inequalities, the necessary optimality conditions in Theorem 3.1 and the stationarity concepts in Definition3.2are typically not very useful in practical applications. In the following Sections4and 5, we will derive multiplier systems that are easier to work with and more suitable as starting points for the development of numerical solution algorithms.
4. Strong stationarity conditions for the problem ( P )
The aim of this section is to establish so-called strong stationarity conditions for the multiobjective optimal control problem (P), i.e., multiplier systems that are equivalent to the purely primal necessary optimality conditions in Definition 3.2. As already pointed out in the introduction, in the single-objective setting, such stationarity systems are well-known for various problem classes. See, e.g., [13, 14, 18, 19, 38, 40, 41] and the references therein for some examples. In the multiobjective context, the situation is different since the additional layer of non-smoothness in (1.1) and the non-differentiability of the control-to-state mappingS create a nested structure that is generally hard to handle analytically. In the following, we will show that, for the problem (P), the difficulties arising from the doubly non-smooth behavior in (1.1) can be resolved by exploiting the self-adjointness property of the operatorS0(u;·)−1in (1.2). The starting point of our investigation is:
Lemma 4.1 (Behavior of higher-order weak derivatives on level sets). Suppose that a functionw∈Wk,1(Ω), k∈N, is given. Then, for everyα∈Nd0 with 1≤ |α| ≤kand every b∈R, it holds∂αw= 0 a.e. in{w=b}.
Proof. We use induction w.r.t. the absolute value l:=|α| of the multi-indexαto establish the claim: Suppose that an arbitrary but fixedk∈Nand aw∈Wk,1(Ω) are given. Then, for everyi= 1, . . . , d, the classical lemma of Stampacchia, see [2], Proposition 5.8.2, implies that (∂iw)1{w=b}= 0 holds as an identity in L2(Ω). This proves the assertion for l= 1. It remains to perform the induction step l7→l+ 1. To this end, let us assume
that a multi-indexα∈Nd0 with|α|=l+ 1≤k, l≥1, is given. Then, we can find multi-indicesβ, γ∈Nd0 with
|β|=l, |γ|= 1, andα=β+γ. From the induction hypothesis, we obtain that∂βwvanishes a.e. on{w=b}, i.e., it holds (∂βw)1{w=b}= 0∈L2(Ω) and, as a consequence,1{w=b}=1{∂βw=0}1{w=b}∈L2(Ω). Using again the classical version of Stampacchia’s lemma, we further obtain that (∂β+γw)1{∂βw=0}= 0∈L2(Ω). Combining the last two identities yields that
(∂αw)1{w=b}= (∂β+γw)1{∂βw=0}1{w=b}= 0∈L2(Ω).
Thus,∂αw= 0 a.e. in{w=b}and the induction step is complete. This proves the claim of the lemma.
We remark that, in a less general format, the above result has already been used in [15], proof of Lemma A.1 and [16], proof of Theorem 2.2. By applying Lemma 4.1 to the PDE (2.3), it is straightforward to check that the directional derivative S0(u;·) indeed satisfies the identity (1.2). To be more precise, we have:
Lemma 4.2(Properties ofS0(u;·) andS0(u;·)−1). Consider an arbitrary but fixed controlu∈L2(Ω)with state y:=S(u). Then, the following is true:
i) The mapS0(u;·) :L2(Ω)→H01(Ω)∩H2(Ω) is bi-Lipschitz and its inverse is given by
S0(u;·)−1:H01(Ω)∩H2(Ω)→L2(Ω), w7→ −∆w+1{y=0}max(0, w) +1{y>0}w. (4.1) ii) The map S0(u;·)−1:H01(Ω)∩H2(Ω) →L2(Ω) admits a unique, globally Lipschitz continuous extension
S0(u;·)−1:L2(Ω)→(H01(Ω)∩H2(Ω))∗, and this extension satisfies S0(u;·)−1(w), z
H01∩H2 = Z
Ω
w(−∆z) +1{y=0}max(0, w)z+1{y>0}wzdx
∀z∈H01(Ω)∩H2(Ω) ∀w∈L2(Ω).
(4.2)
iii) For every z∈H01(Ω)∩H2(Ω), it holds u, S0(u;·)−1(z)
L2 =
−∆u+1{y>0}u, z
H10∩H2. (4.3)
Here, ∆u∈(H01(Ω)∩H2(Ω))∗ denotes the “very weak” Dirichlet Laplacian of the function u∈L2(Ω), i.e., the unique element of(H01(Ω)∩H2(Ω))∗ that satisfies the adjoint relation(u,∆z)L2 =h∆u, ziH1
0∩H2
for allz∈H01(Ω)∩H2(Ω).
iv) For everyz∈H01(Ω)∩H2(Ω), it holds u, S0(u;·)−1(z)
L2 =
S0(u;·)−1(u), z
H10∩H2.
Proof. Parti)of the lemma is a trivial consequence of Proposition2.2. To proveii), we note that the right-hand side of (4.2) defines a globally Lipschitz continuous map fromL2(Ω) to (H01(Ω)∩H2(Ω))∗ that coincides with S0(u;·)−1 onH01(Ω)∩H2(Ω) by Green’s formula. (Recall thatH01(Ω)∩H2(Ω),L2(Ω), and (H01(Ω)∩H2(Ω))∗ are interpreted as a Gelfand triple,i.e.,H01(Ω)∩H2(Ω),→L2(Ω),→(H01(Ω)∩H2(Ω))∗.) The functionS0(u;·)−1 thus admits an extension with the desired properties. SinceH01(Ω)∩H2(Ω) is dense inL2(Ω), we further know that there can only be oneL2-Lipschitz continuous extension ofS0(u;·). This establishesii). It remains to prove iii) andiv). To this end, we note that Lemma4.1and the definition ofS imply
Z
Ω
1{y=0}max(0, z)udx= Z
Ω
1{y=0}max(0, z)(−∆y+ max(0, y))dx= 0
and, analogously, Z
Ω
1{y=0}zmax(0, u)dx= Z
Ω
1{y=0}zmax(0,−∆y+ max(0, y))dx= 0 for allz∈H01(Ω)∩H2(Ω). Combining the above with (4.1) and (4.2) yields
u, S0(u;·)−1(z)
L2 = Z
Ω
u −∆z+1{y=0}max(0, z) +1{y>0}z dx=
Z
Ω
u −∆z+1{y>0}z dx
= Z
Ω
u(−∆z) +1{y=0}max(0, u)z+1{y>0}uzdx=
S0(u;·)−1(u), z
H01∩H2
for allz∈H01(Ω)∩H2(Ω). This establishesiii) andiv)and completes the proof.
Remark 4.3. An argument based on Stampacchia’s lemma similar to that in the proof of Lemma4.2has also been used in [18], Section 5 for the analysis of the solution map of a quasilinear partial differential equation involving a term of the formg(y)∇ywith a piecewise smoothg:R→R. For this PDE, the lemma of Stampacchia for first weak derivatives allows to show that all terms that could possibly prevent the expression g0(y;z)∇y from being linear inzare negligible, and to establish that the solution map of the considered quasilinear partial differential equation is Gˆateaux differentiable in spite of the fact that it contains the non-smooth Nemytskii operator g. Compare also with Theorem6.1 and Remark6.3 in this context. We point out that, for the PDE (2.1), the situation is different since the maps S, S0(u;·), and S0(u;·)−1 are not Gˆateaux differentiable but contain “proper” non-differentiabilities,cf.Proposition2.2iv)and also the example in Section6.
As we will see below, Lemma4.2 makes it possible to handle the non-linearity of the directional derivative S0(u;·) in the necessary optimality conditions (3.4), (3.5), and (3.6). To deal with the multiobjective aspect in these conditions, we need the following infinite-dimensional version of Tucker’s/Motzkin’s theorem of the alternative (cf.[22], Thms. 3.22, 3.24 and also [26], Prop. 2.2):
Lemma 4.4 (Existence of multipliers in general Hilbert spaces). Suppose that V is a real Hilbert space and that w∗1,. . . ,w∗N,N ∈N, are given elements ofV∗. Then, it holds
@z∈V : hw∗n, ziV <0 ∀n= 1, . . . , N
⇐⇒ ∃λ∈RN : λn≥0 ∀n= 1, . . . , N,
N
X
n=1
λn= 1,
N
X
n=1
λnwn∗= 0 (4.4)
and
@z∈V : hw∗n, ziV ≤0 ∀n= 1, . . . , N, hw∗n, ziV <0 for at least onen
⇐⇒ ∃λ∈RN : λn>0 ∀n= 1, . . . , N,
N
X
n=1
λn= 1,
N
X
n=1
λnwn∗= 0. (4.5)
Proof. We begin with (4.4): If we assume that the right-hand side of (4.4) holds and that there exists az∈V withhw∗n, ziV <0 for all n= 1, . . . , N, then we arrive at the contradiction
0 =
* N X
n=1
λnw∗n, z +
V
<0.
This proves “⇐”. To establish “⇒”, we note that the condition on the left-hand side of (4.4) can be recast as max (hw∗1, ziV , . . . ,hw∗N, ziV)≥0 ∀z∈V,
or, equivalently, as 0∈∂c(g◦F)(0), where ∂c denotes the convex subdifferential and where g and F are the functions defined by
g:RN →R, g(x1, . . . , xN) := max(x1, . . . , xN) and F :V →RN, F(z) := (hwn∗, ziV)n=1,...,N. Using the chain rule for the convex subdifferential, see [23], Proposition I-5.7, and the formula for the subdifferential of the vector-maximum ([50], Ex. 8.26), we may now deduce that
0∈F∗∂cg(F(0)) =F∗∂cg(0) =F∗ (
µ∈RN
µn≥0∀n= 1, . . . , N,
N
X
n=1
µn= 1 )
.
This proves the existence of aλ∈RN with the properties on the right-hand side of (4.4) and establishes “⇒”.
It remains to show (4.5). As in the case of (4.4), the implication “⇐” in (4.5) follows immediately by contradiction. To prove “⇒”, we note that, by the Riesz representation theorem, we can find elementswn∈V, n= 1, . . . , N, such thathw∗n,·iV = (wn,·)V holds for all n. Suppose that at least one of these wn is not zero (else the proof is trivial) and denote the subspace spanned by the elementswn with W. Then,W is obviously finite-dimensional and possesses a (·,·)V-orthonormal basis e1, . . . , eM, 1 ≤M ≤N. Let αnm, n = 1, . . . , N, m= 1, . . . , M, be the coordinates ofwn w.r.t. the basis{em}. Then, it holds
* w∗n,
M
X
m=1
βmem
+
V
=
M
X
l=1
αnlel,
M
X
m=1
βmem
!
V
=
M
X
m=1
αnmβm
for allβ∈RM, and we may use the left-hand side of (4.5) to conclude that
@β ∈RM : Aβ∈(−∞,0]N, Aβ6= 0,
where A∈RN×M is the matrix defined byA:= (αnm)n=1,...,N,m=1,...,M. From the classical, finite-dimensional version of Tucker’s theorem of the alternative, see [37], Theorem II-4.3, we may now deduce that there exists a λ∈RN satisfyingATλ= 0,PN
n=1λn= 1, and λn>0 for alln= 1, . . . , N, and from the definition ofAthat
N
X
n=1
λnwn=
M
X
m=1 N
X
n=1
λnαnmem= 0.
This establishes the implication “⇒” in (4.5) and completes the proof.
We are now in the position to prove the main result of this section and the paper as a whole:
Theorem 4.5 (Strong stationarity conditions).
i) A controlu¯∈L2(Ω) with state y¯:=S(¯u)is weakly Pareto stationary for (P) if and only if there exist an adjoint statep¯and a multiplier ¯λsuch thatu,¯ y,¯ p, and¯ ¯λsatisfy the system
¯
u,p¯∈L2(Ω), y¯∈H01(Ω)∩H2(Ω), ¯λ∈RN,
¯λn ≥0 ∀n= 1, . . . , N,
N
X
n=1
λ¯n= 1,
−∆¯y+ max(0,y) = ¯¯ u, −∆¯p+1{¯y>0}p¯=
N
X
n=1
λ¯njn0(¯y), p¯+
N
X
n=1
¯λnνnu¯= 0.
(4.6)
ii) A control u¯∈L2(Ω) with statey¯:=S(¯u)is Pareto stationary for (P) (in the ordinary sense) if and only if there exist an adjoint statep¯and a multiplierλ¯ such thatu,¯ y,¯ p, and¯ ¯λsatisfy the system
¯
u,p¯∈L2(Ω), y¯∈H01(Ω)∩H2(Ω), ¯λ∈RN,
¯λn >0 ∀n= 1, . . . , N,
N
X
n=1
λ¯n= 1,
−∆¯y+ max(0,y) = ¯¯ u, −∆¯p+1{¯y>0}p¯=
N
X
n=1
λ¯njn0(¯y), p¯+
N
X
n=1
¯λnνnu¯= 0.
(4.7)
iii) A controlu¯∈L2(Ω)is Pareto stationary for (P) (in the ordinary sense) if and only if it is properly Pareto stationary for (P).
Proof. We begin with i): From Definition 3.2i), equation (4.3), and Lemma 4.2i), it follows that a control
¯
u∈L2(Ω) with state ¯y:=S(¯u) is weakly Pareto stationary for (P) if and only if
@z∈H01(Ω)∩H2(Ω) : hjn0(¯y), ziH1
0∩H2+νn u, S¯ 0(¯u;·)−1(z)
L2 =
jn0(¯y) +νn −∆¯u+1{¯y>0}u¯ , z
H01∩H2<0 ∀n= 1, . . . , N.
(4.8) Here, the Laplacian ∆¯uis again understood in the very weak sense. Due to Lemma4.4, we further know that (4.8) is equivalent to the statement
∃¯λ∈RN : λ¯n≥0 ∀n= 1, . . . , N,
N
X
n=1
¯λn = 1,
N
X
n=1
¯λn jn0(¯y) +νn −∆¯u+1{¯y>0}u¯
= 0∈ H01(Ω)∩H2(Ω)∗
.
(4.9)
If we now define ¯p:=−PN
n=1¯λnνnu, then the assertion in¯ i)follows immediately. To establishii), we can use exactly the same arguments as fori)(with (4.4) replaced by (4.5)). It remains to proveiii). To this end, let us suppose that ¯uis Pareto stationary in the ordinary sense, that ¯pand ¯λare as in (4.7), and that we are given av∈ L2(Ω) with associated directional derivativez:=S0(¯u;v)∈H01(Ω)∩H2(Ω) such that hjl0(¯y), S0(¯u;v)iH1
0∩H2+ νl(¯u, v)L2<0 holds for somel∈ {1, . . . , N}. Then, Lemma4.2iii)yields
hj0l(¯y), S0(¯u;v)iH1
0∩H2+νl(¯u, v)L2=
jl0(¯y) +νl −∆¯u+1{¯y>0}u¯ , z
H01∩H2,
and we may use the strong stationarity system (4.7) and the same arguments as in (4.9) to deduce that
N
X
n=1
λ¯n
jn0(¯y) +νn −∆¯u+1{¯y>0}u¯ , z
H01∩H2 = 0.
The above implies (completely analogously to (2.7), again by Lemma4.2iii), and since ¯λn>0 for alln) that 0<−
hjl0(¯y), S0(¯u;v)iH1
0∩H2+νl(¯u, v)L2
= 1 λ¯l
X
n6=l
λ¯n
jn0(¯y) +νn −∆¯u+1{¯y>0}u¯ , z
H01∩H2
≤ 1 λ¯l
max
n=1,...,N
jn0(¯y) +νn −∆¯u+1{¯y>0}u¯ , z
H01∩H2
≤
min
n=1,...,N
¯λn
−1
max
n=1,...,N
hjn0(¯y), S0(¯u;v)iH1
0∩H2+νn(¯u, v)L2
.
The proper Pareto stationarity of ¯unow follows immediately, see Definition3.2iii). Since the reverse implication is trivial, this completes the proof.
Several things are noteworthy regarding the last result:
Remark 4.6.
i) The structure of the strong stationarity systems in Theorem4.5is completely analogous to that of classi- cal first-order necessary optimality conditions for smooth, finite-dimensional multiobjective optimization problems, see,e.g., [22], Section 3.3, [39], Section 3.1.1, and [27], Section 4. In particular, (4.6) and (4.7) do not contain any kind of evaluation of the subdifferential of the function max(0,·). Section6 will give an explanation for this behavior.
ii) From [14], Theorem 3.18, it follows straightforwardly that the lines
−∆¯p+1{¯y>0}p¯=
N
X
n=1
¯λnjn0(¯y) and p¯+
N
X
n=1
λ¯nνnu¯= 0
in the strong stationarity conditions (4.6) and (4.7) imply the existence of a generalized derivative G∈
∂BssS(¯u) with
N
X
n=1
¯λn G∗jn0(¯y) +νnu¯
= 0. (4.10)
Here, ∂ssBS(¯u)⊂L(L2(Ω), H01(Ω)∩H2(Ω)) is the strong-strong Bouligand subdifferential of the solution map S associated with (2.1) in the sense of [14], Definition 3.1, andG∗ ∈L((H01(Ω)∩H2(Ω))∗, L2(Ω)) denotes the adjoint of G. Due to the chain rule for the Bouligand subdifferential ∂B and the fact that the Bouligand subdifferential is smaller than the subdifferential of Clarke (which we denote by∂C in the following), (4.10) further yields that
0∈
N
X
n=1
¯λn ∂B(jn◦S)(¯u) +νnu¯
⊂
N
X
n=1
λ¯n∂B Jn(S(·),·) (¯u)⊂
N
X
n=1
¯λn∂C Jn(S(·),·) (¯u).
The strong stationarity conditions (4.6) and (4.7) thus imply that there exist elements of the Clarke sub- differentials∂CJn(S(·),·) of the reduced objective functionsJn(S(·),·),n= 1, . . . , N, of (P) at ¯usuch that a linear combination of these elements with the coefficients ¯λn vanishes. Note that this is precisely the
“standard” first-order necessary optimality condition for non-smooth multiobjective optimization prob- lems, see,e.g., [10, 30,39]. The systems (4.6) and (4.7) are thus more rigorous than ordinary optimality conditions based on Clarke’s generalized differential (and also stronger than conditions which exploit the subdifferential calculus of [14]). This, along with the equivalence to the purely primal stationarity concepts in Definition3.2, justifies calling (4.6) and (4.7)strong stationarity conditions.
iii) Theorem4.5is closely related to classical scalarization approaches (as found,e.g., in [22], Sects. 3 and 4) in the sense that the systems (4.6) and (4.7) can be identified with strong stationarity conditions (in the ordinary, single-objective sense) of optimal control problems governed by (2.1) with objective functions of the formu7→PN
n=1λ¯nJn(S(u), u). Compare, e.g., with [14], Theorem 4.12 in this context. The main insight provided by Theorem4.5is, of course, that every weak, ordinary, or proper stationary point of (P) is a strongly stationary point of a suitably scalarized auxiliary problem.
iv) We would like to point out that the proof of Theorem 4.5relies heavily on the fact that the non-smooth Nemytskii operator max(0,·) in the PDE (2.1) interacts in a special way with the Tikhonov regularization terms in the objective functions of (P) and that the control space of (P) is all ofL2(Ω). If more general non-smooth Nemytskii operators or objective functionsJn:H01(Ω)∩H2(Ω)×L2(Ω)→Rare considered or if the controls are subject to additional constraints, then it is still often possible to establish stationarity systems similar to those in (4.6) and (4.7) by using a scalarization approach and regularization. However, in this general situation, it is typically completely unclear how the resulting optimality conditions are related to the notions of stationarity in Definition3.2(or if they are of a sensible strength at all). Compare, for instance, with the optimality conditions derived by regularization for scalar optimal control problems with general objective functions governed by non-smooth semilinear PDEs in [14], Section 4.1 in this context and also with the analysis of Section5.
As a direct consequence of Theorem 4.5, we obtain:
Corollary 4.7 (Increased regularity of Pareto stationary points).
i) If the mapsjn, n= 1, . . . , N, are continuously differentiable as functions from H01(Ω) toR, i.e., if there exist continuously differentiable ˆjn :H01(Ω) →R such that jn = ˆjn◦EH1
0 holds for all n, where EH1 0 : H01(Ω)∩H2(Ω)→H01(Ω) denotes the canonical embedding of H01(Ω)∩H2(Ω) into H01(Ω), then every Pareto stationary controlu¯ of (P) is an element of H01(Ω).
ii) If the maps jn,n= 1, . . . , N, are continuously differentiable as functions from L2(Ω) toR, i.e., if there exist continuously differentiable ˆjn : L2(Ω) →R such that jn = ˆjn ◦EL2 holds for all n, where EL2 : H01(Ω)∩H2(Ω) → L2(Ω) denotes the canonical embedding of H01(Ω)∩H2(Ω) into L2(Ω), then every Pareto stationary controlu¯ of (P) is an element of H01(Ω)∩H2(Ω).
Proof. From Theorem4.5, we obtain that, for every Pareto stationary ¯u∈L2(Ω) with state ¯y:=S(¯u), we can find a multiplier ¯λ∈RN and an adjoint state ¯p∈L2(Ω) with
¯λn>0 ∀n= 1, . . . , N,
N
X
n=1
λ¯n= 1, p¯+
N
X
n=1
¯λnνnu¯= 0, −∆¯p+1{¯y>0}p¯=
N
X
n=1
¯λnjn0(¯y). (4.11)
If we assume that there exist continuously differentiable functions ˆjn :H01(Ω)→R such that jn = ˆjn◦EH1
0
holds for all n, then it follows from (4.11), the chain rule, and the definitions of the very weak Laplacian and
